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In this paper we introduce and study the separable weakbounded approximation properties which is strictly stronger than the ap-proximation property and but weaker than the bounded approximationproperty. It provides new sufficient conditions for the metric approxima-tion property for a dual Banach space. 1. IntroductionLet X and Y be Banach spaces. We denote by B(X,Y) the space of boundedlinear operators from X into Y, and by F(X,Y), K(X,Y ), W(X,Y), andB S (X,Y) its subspaces of finite rank operators, compact operators, weaklycompact operators, and separable-valued bounded linear operators.Recall that a Banach space X is said to have the approximation property(AP) if there exists a net (S α ) ⊂ F(X,Y ) such that S α → I X uniformly oncompact subsets of X. If (S α ) can be chosen with sup kS α k ≤ 1, then X issaid to have the metric approximation property (MAP). The following is a longstanding open problem [1].The Metric Approximation Problem. Does the approximation propertyof the dual space X

Let q > 1 be an odd integer and c be a fixed integer with(c,q) = 1. For each integer a with 1 ≤ a ≤ q −1, it is clear that thereexists one and only one b with 0 ≤b ≤q −1 such that ab ≡c (mod q).Let N(c,q) denote the number of all solutions of the congruence equationab ≡c (mod q) for 1 ≤a,b ≤q−1 in which a and b are of opposite parity,where b is defined by the congruence equation bb ≡1 (modq). The mainpurpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involvingN(c,q) − 12 φ(q)and Kloosterman sums, and give a sharper asymptoticformula for it. 1. IntroductionLet p be an odd prime and c be a fixed integer with (c,p) = 1. For eachinteger a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one bwith 0 ≤ b ≤ p − 1 such that ab ≡ c (mod p). Let M(c,p) denote the numberof cases in which a and b are of opposite parity. In reference [4], D. H. Lehmerasked to study M(1,p) or at least to say something nontrivial about it. It isknown that M(1,p) ≡ 2 or 0 (mod 4) when p ≡ ±1 (mod 4). For generalodd number q ≥ 3, Wenpeng Zhang [6] studied the asymptotic properties ofM(1,q), and obtained a sharp asymptotic formula:M(1,q) =12φ(q) +O q

This paper is devoted to investigate the singular directionsof meromorphic functions in some angular domains. We will confirm theexistence of Hayman T directions in some angular domains. This is acontinuous work of Yang [8] and Zheng [10]. 1. Introduction and main resultsLet f(z) be a meromorphic function on the whole complex plane. We willuse the standard notation of the Nevanlinna theory of meromorphic functions,such as T(r,f), N(r,f), m(r,f), δ(a,f). For the detail, see [7]. The order andlower order of it are defined as followsλ(f) = limsup r→∞ logT(r,f)lograndµ(f) = liminf r→∞ logT(r,f)logr.In view of the second fundamental theorem of Nevanlinna, Zheng [11] intro-duced a new singular direction, which is named T direction.Definition 1.1. A direction L : argz = θ is called a T direction of f(z) if forany e > 0, we havelimsup r→∞ N(r,Z e (θ),f = a)T(r,f)> 0for all but at most two values of a in the extended complex plane Cb. HereN(r,Ω,f = a) =Z r1 n(t,Ω,f = a)tdt, Received December 23, 2011; Revised May 23, 2014.2010 Mathematics Subject Classification. Primary 30D10; Secondary 30D20, 30B10,34M05.Key words and phrases. Hayman T direction, angular domain, P´olya peaks, order.

We give the characterization of H¨older differentiability pointsand non-differentiability points of the Riesz-N´agy-Taka´cs (RNT) singularfunction Ψ a,p satisfying Ψ a,p (a) = p. It generalizes recent multifractaland metric number theoretical results associated with the RNT function.Besides, we classify the singular functions using the singularity order de-duced from the H¨older derivative giving the information that a strictlyincreasing smooth function having a positive derivative Lebesgue almosteverywhere has the singularity order 1 and the RNT function Ψ a,p hasthe singularity order g(a,p) = alogp+(1−a)log(1−p)aloga+(1−a)log(1−a) ≥1. 1. IntroductionRecently many ([8, 9, 10, 16]) studied the Cantor function, a singular func-tion which is not strictly increasing and the Minkowski’s Question Mark func-tion which is a strictly increasing singular function. Also J. Parad´isetal. ([17])studied some conditions of the null and infinite derivatives of the RNT strictlyincreasing singular function using metric number theory.Recently we ([4]) also studied multifractal characterization of the null andinfinite derivative sets and the non-differentiability set of the RNT singularfunction, which is the typical singular function related to mutifractal theory.In this paper, we employ the Ho¨lder derivative, which is a generalized form ofthe usual derivative, of the RNT function on the unit interval. This definitionextends the concept of the singularity to the general singularity for a strictlyincreasing continuous function. For every point in the unit interval, we ([4])can give some code or dyadic expansion using digit 0 and 1, generating thedistribution set determined by the frequency of the zero in its expansion. Thedistribution sets in the unit interval are the local dimension sets by a self-similar

Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x,y 2 R, the vertices x and y are adjacent if and only if xy 2 S(I).

The intersection graph of a group G is an undirected graphwithout loops and multiple edges defined as follows: the vertex set is theset of all proper non-trivial subgroups of G, and there is an edge betweentwo distinct vertices Hand Kif and only if H∩K6= 1 where 1 denotes thetrivial subgroup of G.In this paper we characterize all finite groups whoseintersection graphs are planar. Our methods are elementary. Among thegraphs similar to the intersection graphs, we may count the subgrouplattice and the subgroup graph of a group, each of whose planarity wasalready considered before in [2, 10, 11, 12]. 1. Introduction and preliminariesA graph is called planar if it can be drawn on the plane in such a way thatits edges intersect only at their endpoints. There are interesting graphs con-structed from algebraic objects such as the subgroup lattice and the subgroupgraph of a group. Planarity of the subgroup lattice and the subgroup graph ofa group were studied by Bohanon and Reid in [2] and by Schmidt in [10, 11]and by Starr and Turner III in [12], and planarity of the intersection graph ofa module over any ring was studied in [13].Here we study planarity of the intersection graph of a finite group. Let G bea group. By the intersection graph of G we mean an undirected graph withoutloops and multiple edges defined as follows: the vertex set is the set of allproper non-trivial subgroups of G, and there is an edge between two distinctvertices H and K if and only if H∩K 6= 1 where 1 denotes the trivial subgroupof G.We call a group planar if its intersection graph is planar. For any naturalnumbers m and n, we use C

It is known that the ranks of the semigroups , and (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on , respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of and are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups , and . Firstly, we characterize the structure of the minimal generating sets of . As applications, we obtain that the number of distinct minimal generating sets is . Secondly, we show that, for , the m-potent ranks of the semigroups and are also n and 2n, respectively. Finally, we find that the 2-potent rank of is n + 1.

Recently, it was proved that every p-Schur ring over an abel-ian group of order p 3 is Schurian. In this paper, we prove that every com-mutative p-Schur ring over a non-abelian group of order p 3 is Schurian. 1. IntroductionLet H be a finite group. We denote by CH the group algebra of H over thecomplex number field C. For a nonempty subset T ⊆ H, we set T :=P t∈T twhich is treated as an element of CH.A subalgebraA ofthe groupalgebraCH iscalled aSchurringoverH if thereexists a partition Bsets(A) := {T 0 ,T 1 ,...,T r } of H satisfying the followingconditions:(i) {T i | T i ∈ Bsets(A)} is a linear basis of A;(ii) T 0 = {1 H };(iii) T − 1i := {t − | t ∈ T i } ∈ Bsets(A) for all T i ∈ Bsets(A).A Schur ring A over a p-group H is called a p-Schur ring if the size of everyelement in Bsets(A) is a power of p, where p is a prime.Let G be a subgroup of Sym(H) containing the left regular representationof H. We denote by T 0 = {1 H }, T 1 ,...,T r the orbits of the stabilizer G 1 H .The transitivity module V (H,G

In 2005 Kadiri proved that the Riemann zeta function ζ(s)does not vanish in the regionRe(s) ≥ 1 −1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 5.69693. In this paper we will show that R 0 can be taken R 0 =5.68371 using Kadiri’smethod together with Platt’s numerical verificationof Riemann Hypothesis. 1. IntroductionIt is well known that the distribution of prime numbers is deeply relatedwith the zeros of the Riemann zeta function ζ(s). Let π(x) be the number ofprimes up to x. Hadamard and de la Vall´ee Poussin proved the prime numbertheorem which states that π(x) is asymptotic to Li(x) =R x2dtlogt . In 1899, dela Vall´ee Poussin proved that ζ(s) does not vanish in the regionRe(s) ≥ 1−1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 34.82. From this zero-free region for ζ(s) the error term π(x)−Li(x)can be estimated:π(x) −Li(x) = O x exp −rlogxR 0 when x → ∞.See [18] and Section 1 of [7]. The constant R 0 has been improved by manyauthors, particularly by Rosser [13, 14, 15], Schoenfeld [14, 15], Stechkin [16],Ford [1, 2], and Kadiri [7]. Recently Kadiri improved significantly on R

A k-hypertournament H on n vertices, where 2 � kn, is a pair H = (V,A), where V is the vertex set of H and A is a set of k-tuples of vertices, called arcs, such that for all subsets SV with |S| = k, A contains exactly one permutation of S as an arc. Recently, Li et al. showed that any strong k-hypertournament H on n vertices, where 3 � kn 2, is vertex-pancyclic, an extension of Moon's theorem for tournaments. In this paper, we prove the following generalization of another of Moon's theorems: If H is a strong k-hypertournament on n vertices, where 3 � kn 2, and C is a Hamiltonian cycle in H, then C contains at least three pancyclic arcs.

We consider a condition under which the projectivization P (Ek) of a complex k-bundle Ek → M over an even-dimensional manifold M can have the hard Lefschetz property, affected by [4]. It depends strongly on the rank k of the bundle Ek. Our approach is purely algebraic by using rational Sullivan minimal models [2]. We will give some examples. The contents of this paper is according to [6]. A Poincare duality space Y of the formal dimension fd(Y ) = max{i;H (Y ;Q) = 0} = 2m is said to be cohomologically symplectic (c-symplectic) if u = 0 for some u ∈ H(Y ;Q) and, furthermore, is said to have the hard Lefschetz property (or simply the Lefschetz property) with respect to the c-symplectic class u, if the maps ∪u : Hm−j(Y ;Q) → H(Y ;Q) 0 ≤ j ≤ m are monomorphisms (then called the Lefschetz maps) [7]. (Then we often say that H∗(Y ;Q) is a Lefschetz algebra [4].) For example, a compact Kahler manifold has the hard Lefschetz property [7], [3, Theorem 4.35]. Let M be an even-dimensional manifold and ξ : E → M be a complex k-bundle over M . The projectivization of the bundle ξ P (ξ) : CP k−1 j → P (E) → M satisfies the rational cohomology algebra condition (∗) : H∗(P (E);Q) = H∗(M ;Q)[x] (xk + c1xk−1 + · · ·+ cjxk−j + · · ·+ ck−1x+ ck) where ci are the Chern classes of ξ and x is a degree 2 class generating the cohomology of the complex projective space fiber (Leray-Hirsch theorem) [1], [4], [7, p.122]. The manifold P (E) appears as the exceptional divisor in blow-up construction for a certain embedding of M [5], [7, Chap.4]. When M is a non-toral symplectic nilmanifold of dimension 2n, there is a bundle E such that P (E) is not Lefschetz [8], [4, Example 4.4]. In general, for a 2k-dimensional manifold M and a fibration

In this paper, we consider characters of SL 2 (Z) and thenapply them to newforms of integral weight, level 4 and of trivial character.More precisely, we prove that all of them are actually level 1 forms ofsome nontrivial character. As a byproduct, we prove that they all areeigenfunctions of the Fricke involution with eigenvalue −1. IntroductionThe Fricke involution W N of level N, also known as the canonical involution,actsonthe spaceofnewformsoflevelN, integralweightk, and trivialcharacter.Here k is necessarily even and positive. It is well-known that Hecke eigenformsbehave well under the Fricke involution. More specifically, if f is a normalizedHecke eigenform of some level N, weight k and of trivial character, then wehave f| k W N = cg with c ∈ C × and g another normalized Hecke eigenform inthe same space (see Lemma 1.1 below or Theorem 4.6.16 in [6]). The Fouriercoefficients of g can be explicitly determined by that of f but the scalar c isleft mysterious in general.Question 1. Can we explicitly determine c with the information on f?Let f =P

In this paper we study the sequences defined by the last and the last non-zero digits of in base b. For the sequence given by the last digits of in base b, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last non-zero digits of in base b (which had been studied only for b = 10) we show the non-periodicity of the sequence when b is an odd prime power and when it is even and square-free. We also show that if the sequence is periodic and conjecture that this is the only such case.

In this paper, we study some quantity equivalent to the normof Bloch to A pα composition operator where A pα is the weighted Bergmanspace on the unit ball of C n (0 < p < ∞ and −1 < α < ∞). 1. IntroductionLet n ≥ 1 be an integer. Let H(B) denote the space of all holomorphicfunctions in the unit ball B ≡ B n of the complex n-dimensional Euclideanspace C n . Let U stand for B 1 . The Bloch space B = B(U) is defined byB = {f ∈ H(U) : sup z∈U |f ′ (z)|(1 − |z| 2 ) < ∞}. With the norm kfk B =|f(0)|+sup z∈U |f ′ (z)|(1−|z| 2 ), B is a Banach space. Let ν denote the Lebesguemeasure on C n , so normalized that ν(B) = 1. For −1 < α < ∞, we setc α = Γ(n+α+1)/{Γ(n+1)Γ(α+1)} and dν α (z) = c α (1−|z| 2 ) α dν(z), z ∈ B.Note that ν α (B) = 1. For 0 < p < ∞ and −1 < α < ∞, the weighted Bergmanspace A pα ≡ A pα (B) is defined by A pα = {f ∈ H(B) :R B |f(z)| dν (z) < ∞}.For f ∈ A pα , we write kfk A pα =R B |f(z)| p dν α (z) 1p .In 2003, Kwon and Lee proved the following theorem. This result relates tothe pull-back property (see [1], [2], [5], [7] for example).Theorem 1 ([6]). Let f : U → U be a holomorphic function with f(0) = 0.For 1 ≤ p < ∞ and −1 < α < ∞, the bounded operator C

Let R be a commutative ring with 1 � . In this paper, we introduce the concept of 2-absorbing primary ideal which is a general- ization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I ,t henab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

A finite group G is called a SB-group if every subgroup ofG is either s-quasinormal or abnormal in G. In this paper, we give acomplete classification of those groups which are not SB-groups but allof whose proper subgroups are SB-groups. 1. IntroductionAll groups considered here are assumed to be finite, and all notations em-ployed are standard.Let Σ be an abstract group theoretical property. If all proper subgroups ofa group G have the property Σ but G does not have the property Σ, then Gis called a minimal non-Σ-group. One of the hottest topics in group theory isto determinate the structure of minimal non-Σ-groups and many meaningfulresults about this topic have been obtained. For example, O. J. Schmidt [12]determined the structure of minimal non-nilpotent groups, K. Doerk [4] eluci-dated the structure of minimal non-supersolvable groups, and J. G. Thompson[14] gavethe classificationofnonsolvable groupsall of whoselocal subgroupsaresolvable. These achievements have greatly pushed forward the developments ofgroup theory. The more new results can be referred to refs. [7, 8, 9, 11, 13, 17].The aim of the present article is to study the structures of some minimal non-Σ-groups. Recall that a subgroup H of a group G is an abnormal subgroupif x ∈ hH,H

Let On and POn denote the order-preserving transformation and the partial order-preserving transformation semigroups on the set Xn = {1,... n}, respectively. Then the strictly partial order-preserving transformation semigroup SPOn on the set Xn, under its natural order, is defined by SPOn = POn \ On. In this paper we find necessary and sufficient conditions for any subset of SPO(n, r) to be a (minimal) generating set of SPO(n, r) for 2 ? r ? n-1. © 2014 Korean Mathematical Society.

In the paper, we define a class of convex bodies in R n –parallelsections homothety bodies, and for some special parallel sections homoth-ety bodies, we prove that n-cubes have the minimal Mahler volume. 1. IntroductionThe well-known Mahler’s conjecture (see, e.g., [8], [15], [25] for references)states that, for any origin-symmetric convex body K in R n ,P(K) ≥ P(C n ) =4 n n!(1.1) ,where C n is an n-cube and P(K) = Vol(K)Vol(K ∗ ), which is known as theMahler volume of K.For n = 2, Mahler [16] himself proved the conjecture, and in 1986 Reisner[22] showed that equality holds only for parallelograms. For n = 2, a newproof of inequality (1.1) was obtained by Campi and Gronchi [4]. Recently,Lin and Leng [12] gave a new and intuitive proof of the inequality (1.1) inR 2 . Reisner (see, e.g., [9, 21, 22]) established the same inequality for a class ofbodies that have a high degree of symmetry, known as zonoids. Inequality (1.1)was established by Saint Raymond [24] for bodies which are symmetric withrespect to the coordinate hyperplanes. For the case of polytopes with at most2n+2 vertices (or facets) (see, e.g., [2] for references), Lopez and Reisner [13]proved the inequality (1.1) for n ≤ 8 and the minimal bodies are characterized.Recently, Nazarov, Petrov, Ryabogin and Zvavitch [20] proved that the cube isa strict local minimizer for the Mahler volume in the class of origin-symmetricconvex bodies endowed with the Banach-Mazur distance.

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x,y ∈ X (x 6 y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by �(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(�(G)) is abelian if and only if |G| ≤ 2; |Aut(�(G))| is of prime power if and only if |G| ≤ 2, and |Aut(�(G))| is square-free if and only if |G| ≤ 3. Some new graphs that are useful in studying the automorphism group of �(G) are presented and their main properties are investigated.

Constacyclic codes of length p s over R = Fpm + uFpm + u2Fpm are precisely the ideals of the ring R(x) hxp s −1i . In this paper, we investigate constacyclic codes of length p s over R. The units of the ring R are of the forms , �+u�, �+u� +u 2 and �+u 2 , where �, � and are nonzero elements of Fpm. We obtain the structures and Hamming distances of all (�+u�)-constacyclic codes and (�+u�+u 2 )-constacyclic codes of length p s over R. Furthermore, we classify all cyclic codes of length p s over R, and by using the ring isomorphism we characterize -constacyclic codes of length p s over R.

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring proper- ties pass to power series rings and polynomial rings. It is also shown that �-regular rings are strongly �-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommuta- tive rings appropriate for the situations occurred naturally in studying standard ring theoretic properties. 1. Definitions and notations Throughout this paper, R denotes an associative ring without identity, unless otherwise stated. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). We let eij denote the usual matrix units with 1 in the (i,j)-position and zeros elsewhere, if the base ring has identity 1. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). Zn denotes the ring of integers modulo n. GF(p n ) denotes the Galois field of order p n . J(R) denotes the Jacobson radical of R. | | denotes the cardinality. The characteristic of R is denoted by charR, and |a| denotes the order of a ∈ R in the additive subgroup of R generated by a. R + means the additive Abelian group (R,+). The polynomial ring with an indeterminate x over R is denoted by R(x). While speaking about minimal ring in a certain class of rings, we refer to a ring with minimal order for rings in that class, due to Xue (21). The notation (S) stands for the two-sided ideal of R generated by ∅ 6 S ⊆ R, and we also write (a1,...,an) in place of (S) for simplicity when S = {a1,...,an}. A ring is called Abelian if every idempotent is central. The zero in a nil ring is the only idempotent and so every nil ring is Abelian. The class of Abelian

Let M be a module over a commutative ring R, and let T(M)be its set of torsion elements. The total torsion element graph of Mover R is the graph T(Γ(M)) with vertices all elements of M, and twodistinct vertices m and n are adjacent if and only if m+ n ∈ T(M).In this paper, we study the basic properties and possible structures oftwo (induced) subgraphs Tor 0 (Γ(M)) and T 0 (Γ(M)) of T(Γ(M)), withvertices T(M) \{0}and M\{0}, respectively. The main purpose of thispaper is to extend the definitions and some results given in [6] to a moregeneral total torsion element graph case. 1. IntroductionThe concept of the graph of the zero-divisors of a ring was first introducedby Beck in [12] when discussing the coloring of a commutative ring. For thevertices of the graph, he takes all elements of a commutative ring R and twodistinct vertices a,b ∈ R are adjacent if ab = 0. There are many ways toassociate a graph to a given ring R. The most well-known is certainly thezero-divisor graph Γ(R) introduced in [9] whose vertices are the nonzero zero-divisors of R. Some properties of this graph may be found in [5] and [10]. In [8],Anderson and Badawi define, for a commutative ring R with nonzero identity,its total graph Γ(R). The set of vertices of this graph is R and two differentelements x,y ∈ R are adjacent if and only if x + y ∈ Z(R) which Z(R) is theset of all zero-divisors of R. For a recent generalization of this type of graphsee [7] and [11]. Let M be a module over a commutative ring R and let T(M)be the set of all torsion elements of M. In [14], the notion of the total torsionelement graph of a module over a commutative ring is introduced and denotedby T(Γ(M)), as the graph with all elements of M as vertices and for distinctm,n ∈ M, the vertices m and n are adjacent if and only if m+n ∈ T(M). Theycharacterize the girths and diameters of T(Γ(M)) and two (induced) subgraphs

Let S 0 be the class of normalized univalent harmonic map- pings in the unit disk. A subclass V H (k) of S 0 , whose analytic part is function with bounded boundary rotation, is introduced. Some bounds for functionals, specially harmonic pre-Schwarzian derivative, described in V H (k) are given. where h and g are analytic in , with g(z0) = 0 for some prescribed point z0 ∈ . We call h and g analytic and co-analytic parts of f, respectively. If f is (locally) injective, then f is called (locally) univalent. The Jacobian and second complex dilatation of f are given by Jf(z) = |fz| 2 −|f¯| 2 = |h ' (z)| 2 −|g ' (z)| 2 and !(z) = g ' (z)/h ' (z) (z ∈ ), respectively. A result of Lewy (18) states that f is locally univalent if and only if its Jacobian is never zero, and is sense-preserving if the Jacobian is positive. The sense-preserving case implies |!(z)| < 1 in D. Throughout this paper we will assume that f is locally univalent, sense- preserving, and = D ⊂ C, with z0 = 0, where D is the open unit disk on the complex plane. Following Clunie and Sheil-Small notation (6), the class of all sense-preserving univalent harmonic mappings of D with h(0) = g(0) = h ' (0)−1 = 0 we denote SH, and its subclass for which g ' (0) = 0 by S 0 H. Several fundamental information about harmonic mappings in the plane can be found in e.g. (8). We note that each f satisfying (1.1) in D is uniquely determined by

Nielsen and Rege-Chhawchharia called a ring R right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, there exists a nonzero element r 2 R with f(x)r = 0. Hong et al. called a ring R strongly right McCoy if given nonzero polynomials f(x),g(x) over R with f(x)g(x) = 0, f(x)r = 0 for some nonzero r in the right ideal of R generated by the coefficients ofg(x). Subsequently, Kim et al. observed similar conditions on linear polynomials by finding nonzero r's in various kinds of one-sided ideals generated by coefficients. But almost all results obtained by Kim et al. are concerned with the case of products of linear polynomials. In this paper we examine the nonzero annihilators in the products of general polynomials. A ring is usually called reduced if it has no nonzero nilpotent elements. Cohn (5) called a ring R reversible if ab = 0 implies ba = 0 for a,b ∈ R. Due to Nar- bonne (21), a ring R is called semicommutative if ab = 0 implies aRb = 0 for a,b ∈ R. Reduced rings are reversible and reversible rings are semicommuta- tive, but not conversely in general. Rege and Chhawchharia called R an Ar- mendariz ring (24, Definition 1.1) if whenever any polynomials f(x),g(x) ∈ R(x) satisfy f(x)g(x) = 0, then ab = 0 for each coefficienta of f(x) and b of g(x). Any reduced ring is Armendariz by (3, Lemma 1), but the class of semicom- mutative rings and the class of Armendariz rings don't imply each other by (24, Example 3.2) and (8, Example 14). McCoy (20) showed that if two poly- nomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. In (10), Weiner showed this fact fails in noncommutative rings. Based on this result, Nielsen (22) and Rege and Chhawchharia (24) called a noncommutative ring R right McCoy (resp., left McCoy) if whenever any nonzero polynomials f(x),g(x) ∈ R(x) satisfy f(x)g(x) = 0, then f(x)c = 0 (resp., cg(x) = 0) for some nonzero c ∈ R

A three-valued function f defined on the vertices of a di-graph D = (V,A), f : V → {−1,0,+1} is a minus total dominatingfunction(MTDF) if f(N − (v)) ≥ 1 for each vertex v ∈ V. The minustotal domination number of a digraph D equals the minimum weight ofan MTDF of D. In this paper, we discuss some properties of the minustotal domination number and obtain a few lower bounds of the minustotal domination number on a digraph D. 1. IntroductionFor terminology and notation on graph theory not given here we follow [1].Let G = (V,E) be a simple graph (digraph or undirected graph). Let v be avertex in V . The open neighborhood of v is the set N(v) = {u ∈ V |uv ∈ E}and the degree of v is d G (v) = |N(v)|. The maximum degree and minimumdegree of G are denoted by ∆(G) and δ(G), respectively. When no ambiguitycan occur, we often simply write d(v), ∆, and δ instead of d G (v), ∆(G), andδ(G), respectively. For any S ⊆ V, G[S] is the subgraph induced by S. Wedenote χ(G) by the chromatic number of G. Let D = (V,A) be a digraph. Foreach vertex v ∈ V , let N

In this paper, we first introduce a generalized concept of Berge strong equilibrium for a generalized game G = (Xi;Ti,fi)i∈I of normal form, and using a fixed point theorem for compact acyclic maps in admissible convex sets, we establish the existence theorem of generalized Berge strong equilibrium for the game G with acyclic values. Also, we have demonstrated by examples that our new approach is useful to produce generalized Berge strong equilibria.

In this paper, under some suitable integrability and smooth-ness conditions on f, we establish the central limit theorems forX k≤N k −1 f(S k /σ√k),where S k is the partial sums of strictly stationary mixing random vari-ables with EX 1 = 0 and σ 2 = EX 21 +2P ∞k=1 EX 1 X 1+k . We also estab-lish an almost sure limit behaviors of the above sums. 1. IntroductionLet {X n ,n≥ 1} be a sequence of random variables on some probabilityspace (Ω,F,P). Let F ba denote the σ-field generated by the random variablesX a ,X a+1 ,...,X b . For any two σ-field A,B ⊂ F putρ(A,B) = supˆCov(X,Y)k Xk 2 k Y k 2 : X∈ L 2 (A),Y ∈ L 2 (B)˙,φ(A,B) = sup{| P(B| A) −P(B) |: A∈ A,B∈ B},α(A,B) = sup{| P(AB) −P(A)P(B) |: A∈ A,B∈ B},where and in the sequel k X k p = (E|X| p ) 1/p for 1 ≤ p

【In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces $L^2(\mathbb{N},\mathcal{K})$ (or $L^2(\mathbb{Z},\mathcal{K})$ ) with weight sequence { $A_n$ } of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$ . Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.】

Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim for for and >0. This shows that> is a finite set for . Also, we prove that if is M-sequences in dimension > k and are some positive integers. Here, for a subset T of Spec(R), set .

The aim of this paper is to prove the next result. Let n > 1be an integer and let R be a n!-torsion free semiprime ring. Suppose thatf : R →R is an additive mapping satisfying the relation [f(x),x n ] = 0for all x ∈R. Then f is commuting on R. 1. Introduction and the maintheoremThroughout, R will represent an associative ring with a center Z(R). Letn > 1 be an integer. A ring R is n-torsion free if nx = 0, x ∈ R, impliesx = 0. The Lie product (or a commutator) of elements x,y ∈ R will be denotedby [x,y] (i.e., [x,y] = xy − yx). Recall that a ring R is prime if aRb = {0},a,b ∈ R, implies that either a = 0 or b = 0. Furthermore, a ring R is calledsemiprime if aRa = {0}, a ∈ R, implies a = 0. We will denote by C and Q theextended centroid and the maximal right ring of quotients of a semiprime ringR, respectively. For the explanation of the extended centroid as well as themaximal right ring of quotients of a semiprime ring we refer the reader to [4].As usual, the socle of a ring R will be denoted by soc(R).An additive mapping D : R → R is called a derivation on R if D(xy) =D(x)y + xD(y) holds for all pairs x,y ∈ R. An additive mapping f : R →R is called centralizing on R if [f(x),x] ∈ Z(R) holds for all x ∈ R. In aspecial case, when [f(x),x] = 0 for all x ∈ R, the mapping f is said to becommuting on R. A classical result of Posner [21] (Posner’s second theorem)states that the existence of a nonzero centralizing derivation on a prime ringforces the ring to be commutative. Posner’s second theorem in general cannotbe proved for semiprime rings as shows the following example. Let R

In this paper, a boundary version of the Schwarz and Carath-´eodory inequality are investigated. New inequalities of the Carath´eodory’sinequality and Schwarz lemma at boundary are obtained by taking intoaccount zeros of f(z) function which are different from zero. The sharp-ness of these inequalities is also proved. 1. IntroductionLet f be a function which is holomorphic on the D : {z : |z| < 1} and vanishat z = 0, and suppose that |f| < 1 for all z ∈ D. Then the inequality(1.1) |f(z)| ≤ |z|holds for all z ∈ D, and moreover(1.2) |f ′ (0)| ≤ 1.Equality is achieved in (1.1) (for some nonzero z ∈ D) or in (1.2) if and onlyif f is an entire linear function of the form f(z) = e iα z, where α is a realnumber([2], p. 381).Let the zeros of f be z 1 ,z 2 ,...,z n . If we apply inequality (1.1) to the func-tion f(z)Q nk=1 h 1−z k zz+z k i, we can conclude in the following Schwarz’s inequality:(1.3) |f(z)| ≤ |z|Y nk=1 z −z k 1−z k z and|f ′ (0)| ≤Y nk=1 |z k |. Received May 3, 2013.2010 Mathematics Subject Classification. Primary 30C80.Key words and phrases. Schwarz lemma on the boundary, holomorphic function, Julia-Wolff-Lemma.

J. Malkevitch dened the coded path in r-valent polytopal graphs of uniform face structure and showed many interesting properties of the coded paths. In this paper, we study the simplicity of coded paths in an m-valent planar multigraph which is not a polytopal graph. and theorems briey. A graph G is an ordered triple (V (G), E(G), G) consisting of a nonempty set V (G) of vertices, a set E(G) of edges, and an incidence function G that associates with each edge of G an unordered pair of (not necessarily distinct) elements ofV (G). Ife is an edge andu andv are vertices such that G(e) = uv, then e is said to join two vertices u and v; the vertices u and v are called the endpoints (or endvertices) of the edge e. The endpoints of an edge are said to be incident to an edge and two vertices which are incident with the same edge, are said to be adjacent. Two or more edges that join the same pair of distinct vertices are called parallel edges (or multiple edges). An edge joining a vertex to itself is called a loop. A graph with no loops or no parallel edges is called a simple graph. A graph which is not simple is said to be a multigraph. The number of edges at the vertex v is called the valence of v (or the degree of v) and is denoted by d(v). If every vertex of a graph G has the same valence r, then G is called an r-valent (or r-regular) graph. A graphG is called planar if it can be drawn in the plane so that the edges of the graph intersect only at vertices. When a connected planar graph is drawn in the plane, the regions bounded by edges of the graph which do not contain neither vertices nor edges in their interiors are called faces. There will always be precisely one face which is unbounded in a planar graph and this will be called the innite face . The edges bounding a face are called its sides.

In this paper we investigate (n+1)(n ≥ 3)-dimensional con-tact CR-submanifolds M of (n−1) contact CR-dimension in a completesimply connected Sasakian space form of constant φ-holomorphic sec-tional curvature c 6= −3 which satisfy the condition h(FX,Y )+h(X,FY )= 0 for any vector fields X,Y tangent to M, where h and F denote thesecond fundamental form and a skew-symmetric endomorphism (definedby (2.3)) acting on tangent space of M, respectively. 1. IntroductionLet S 2m+1 be a (2m + 1)-unit sphere in the complex (m + 1)-space C m+1 ,i.e.,S 2m+1 := {(z 1 ,...,z m+1 ) ∈ C m+1 | m X +1j=1 |z j | 2 = 1}.For any point z ∈ S 2m+1 we put ξ = Jz, where J denotes the complex structureof C m+1 . Denoting by π the orthogonal projection : T z C m+1 → T z S 2m+1 andputting φ = π◦J, we can see that the set (φ,ξ,η,g) defines a Sasakian structureon S 2m+1 , where g is the standard metric on S 2m+1 induced from that of C m+1 and η is a 1-form dual to ξ. Hence S 2m+1 can be considered as a Sasakianmanifold of constant curvature 1 (cf. [2, 4, 5, 6, 7, 8, 9]).Let M be an (n+1)-dimensional submanifold tangent to the structure vectorfield ξ of S

The term rank of a matrix A over a semiring S is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to term rank inequalities of matrices over nonbinary Boolean semirings.

In this paper we consider the existence of smooth conics ona general hypersurface of degree din P n . 1. IntroductionLet X be a generalhypersurfaceof degreed in a complex projective space P n .Recall that R e (X) is the open subscheme of the Hilbert scheme Hilb et+1 (X)which parameterizes smooth rational curves of degree e lying on X. Recently J.Harris, M. Roth, and J. Starr obtained the following general result for d 2 and d 2n − 3, then R 1 (X) is empty. Ifd ≤ 2n−3,thenR 1 (X) issmoothofdimension2n−3−d.In this paperwe will investigatethe nonemptynessandsmoothness ofR 2 (X).Theorem 1.1. LetX beageneralhypersurfaceofdegreed inP

Let R be a commutative Noetherian ring, I an ideal of R,M and N two R-modules. We characterize the least integer i such thatH iI (M,N) is not weakly Artinian by using the notion of weakly filterregular sequences. Also, a local-global principle for minimax generalizedlocal cohomology modules is shown and the result generalizes the corre-sponding result for local cohomology modules. 1. IntroductionThroughout this paper, let R be a commutative Noetherian ring, I a properideal of R, M and N two R-modules. The generalized local cohomologymoduleH iI (M,N) = lim−→ n∈N Ext iR (M/I n M,N)was introduced by Herzog in [7]. It is clear that H iI (R,N) is just the ordinarylocal cohomology module H iI (N). For the details about local cohomology, werefer the reader to the book [3].In [6, Definition 3.1], Hajikarimi gave the definition of weakly filter regularsequences in the case that R is a local ring. In Section 2, we generalize it intothe generalunnecessary local case. If M is a finitely generated R-module and Nis a weakly Laskerian R-module, we prove that inf {i | H

InthispaperweestablishsomenewresultsonsumsofHilbertspace frames and Riesz bases. We also provide a correction to somerecentlyestablishedresultsin[2]. 1. IntroductionThroughout this paper, H denotes a separable Hilbert space with the innerproduct h·,· . Recall that a sequence {f i } i∈I ⊆ H is a frame for H if there exist0 < A 6 B < ∞ such that(1.1) Akfk 2 6X i∈I | f,f i i| 2 6 Bkfk 2 for all f ∈ H. The constants A and B are called a lower and upper framebounds, respectively.We call a sequence {f i } i∈I ⊆ H a Bessel sequence for H, if the right handinequality in (1.1) holds for all f ∈ H.Let {f i } i∈I be a Bessel sequence for H. Then the bounded operatorT : H → l 2 , Tf = {hf,f i i} i∈I is called the analysisoperator of {f i } i∈I and its adjointT ∗ : l 2 → H, T ∗ ({c i } i∈I ) =X i∈I c i f i ,is called the synthesisoperator of {f i } i∈I . If {f i } i∈I is a frame for H, the frameoperator for {f i } i∈I is the operator S : H → H given by Sf =P i∈I hf,f i if i .It is clear that hSf,fi =P

Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an Boolean matrix A is the smallest positive integer q such that for some positive integer r and every nonnegative integer i, where denotes the transpose of A. In this paper, we study the upper bound of the competition index of a Boolean matrix. Using the concept of Boolean rank, we determine the upper bound of the competition index of a nearly reducible Boolean matrix.

Let G = (V,E) be a graph. A function f : V → {−1,+1}defined on the vertices of G is a signed total dominating function if thesum of its function values over any open neighborhood is at least one.The signed total domination number of G, γ st (G), is the minimum weightof a signed total dominating function of G. In this paper, we study thesigned total domination number of generalized Petersen graphs P(n,2)and prove that for any integer n ≥ 6, γ st (P(n,2)) = 2⌊ n3 ⌋ + 2t, wheret ≡ n(mod 3) and 0 ≤ t ≤ 2. 1. IntroductionFor notation and graph theory terminology we in general follow [5]. Spe-cially, let G be a graph with vertex set V (G) and edge set E(G). Let v be a ver-tex in V (G). The openneighborhood of v is N G (v) = {u ∈ V (G) | uv ∈ E(G)}.If the graph G is clear from the text, we simply write V , E and N(v) ratherthan V (G), E(G) and N G (v). If X ⊆ V , then hXi is the subgraph induced byX. The distance d(x,y) between two vertices x and y in G is the length of theshortest path from x to y. For a real-valued function f : V → R, the weight off is w(f) =P

In this paper, we prove that the total domination number of a P6-free graph of order n � 3 and minimum degree at least one which is not the cycle of length 6 is at most n+1 , and the bound is sharp. 1. Introduction A total dominating set of a graph G with no isolated vertex is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total dom- ination number of G, denoted by t(G), is the minimum cardinality of a total dominating set of G. Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi (3). For notation and graph theory terminology we in general follow (3). Let G = (V,E) be a graph with vertex set V of order n. The degree, neighborhood and closed neighborhood of a vertex v in the graph G are denoted by d(v), N(v) and N(v) = N(v)∪{v}, respectively. The minimum degree and maximum degree of the graph G are denoted by �(G) and �(G), respectively. For any S ⊆ V , N(S) = S v∈S N(v). Let G(S) denote the graph induced by S. Let Cn, Pn and K1,n−1 denote the cycle, the path and star of order n, respectively. A graph is Pn-free if it does not contain Pn as an induced subgraph.

In this paper we give an explicit formula for the total number of subgroups of a finite abelian -group up to rank three.

In this paper, a class of set-valued mappings is introduced, which is called uniformly same-order. For this sort of mappings, some minimax problems, in which the minimization and the maximization of set-valued mappings are taken in the sense of vector optimization, are investigated without any hypotheses of convexity.

This paper generalizes the Aleksandrov problem and MazurUlam theorem to the case of n-normed spaces. For real n-normed spacesX and Y, we will prove that f is an affine isometry when the mappingsatisfies the weaker assumptions that preserves unit distance, n-colinearand 2-colinear on same-order. 1. IntroductionLet X and Y be metric spaces. A mapping f : X → Y is called an isometryif f satisfies d Y (f(x),f(y)) = d X (x,y) for all x,y ∈ X, where d X (·,·) andd Y (·,·) denote the metrics in the spaces X and Y , respectively. For some fixednumber r > 0, assume that f preserves the distance r, i.e., for all x,y ∈ X withd X (x,y) = r, it holds that d Y (f(x),f(y)) = r. Then r is called a conservative(or preserved) distance for the mapping f.In 1970, Aleksandrov [1] posed the following problem: Examine whether theexistence of a single conservative distance for some mapping T implies that Tis an isometry.In 1932, Mazur and Ulam [8] proved a theorem that every isometry of a realnormed space onto a real normed space is a linear mapping up to a translation.In 1993, Rassias and Semrl [10] proved a series of results on the DOPPˇ(distance one preserving property) for the normed spaces (see also [6, 7, 9]).Since 2004, the Aleksandrovproblem has been discussedin the n-normed spaces(n ≥ 2) (see [2, 3, 4, 5]). Chu, Lee and Park proved:Theorem 1.1 ([3]). Let X and Y be two real linear n-normed spaces. If amapping f : X → Y satisfies the following conditions:(1) f has the n-DOPP;(2) f is n-Lipschitz

In this paper, we show that tree products of certain subgroup separable groups amalgamating normal subgroups are cyclic subgroup separable. We then extend this result to certain graph product of certain subgroup separable groups amalgamating normal subgroups, that is we show that if the graph has exactly one cycle and the cycle is of length at least four, then the graph product is cyclic subgroup separable.

Let Rbe a prime ring, I a nonzero ideal of R, da derivationof R, m(≥ 1),n(≥ 1) two fixed integers and a ∈ R. (i) If a((d(x)y +xd(y) + d(y)x+ yd(x)) n − (xy+ yx)) m = 0 for all x,y ∈ I, then eithera = 0 or R is commutative; (ii) If char(R) 6= 2 and a((d(x)y+ xd(y) +d(y)x+yd(x)) n −(xy+yx)) ∈ Z(R) for all x,y∈ I, then either a= 0 orRis commutative. 1. IntroductionThroughout this paper R always denotes a prime ring with center Z(R),extended centroid C and Q its two-sided Martindale quotient ring. Recall thata ring R is said to be prime, if for any a,b ∈ R, aRb = (0) implies either a = 0or b = 0 and it is semiprime if for any a ∈ R, aRa = (0) implies a = 0. Forany x,y ∈ R, the Lie commutator of x,y is denoted by [x,y] and defined by[x,y] = xy−yx and the anti-commutator is denoted by x◦y and is defined byx ◦ y = xy + yx. By d we mean a derivation of R. A derivation d is inner ifthere exists b ∈ R such that d(x) = [b,x] holds for all x ∈ R.A well known theorem of Posner [15] states that if R is prime and the com-mutator [d(x),x] ∈ Z(R) for all x ∈ R, then either d = 0 or R is commutative.This result of Posner was generalized in many directions by several authorsand they studied the relationship between the structure of prime or semiprimering and the behaviour of additive maps satisfying various conditions. Someauthors have studied the derivations with annihilator conditions in prime andsemiprime rings (see [3], [4], [6], [7], [8], [9]; where further references can befound).In [2], Ashraf and Rehman proved that if R is a prime ring, I is a nonzeroideal of R and d is a derivation of R such that d(x)y+xd(y)+d(y)x+yd(x) =xy + yx for all x,y ∈ I, then R is commutative. Recently, in [1]; Argac andInceboz generalized the above result as follows

It is known that being hierarchical is a necessary and suffi- cient condition for a poset to admit MacWilliams identity. In this paper, we completely characterize the structures of posets which have an associ- ation scheme structure whose relations are indexed by the poset distance between the points in the space. We also derive an explicit formula for the eigenmatrices of association schemes induced by such posets. By using the result of Delsarte which generalizes the MacWilliams identity for linear codes, we give a new proof of the MacWilliams identity for hierarchical linear poset codes.

In this paper, we consider several convolution sums, namely, Ai(m;n; N) (i = 1; 2; 3; 4), Bj(m;n;N) (j = 1; 2; 3), and Ck(m;n;N) (k = 1; 2; 3;:::, 12), and establish certain identities involving their nite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity in- volving the Weierstrass }-function, its derivative and certain linear com- bination of Eisenstein series is established.

In the paper, we have two purposes. Firstly, we estimate thehyper-order of an entire function which shares two functions with it???s ???rstderivative, and two examples are given to show the conclusion is sharp.Secondly, we generalize the Bru??ck conjecture with the idea of sharingfunctions. 1. Introduction and main resultsIn the Nevanlinna theory, the order and the hyper-order of a meromorphicfunction are two important concepts. So, it is meaningful to discuss the prop-erties of the order and the hyper-order for a meromorphic function. Let usrecall the de???nitions of the order and hyper-order of a meromorphic functionf, which are respectively de???ned as (see [12])??(f) = limsup r?????? logT(r,f)logr= limsup r?????? loglogM(r,f)logr,??(f) = limsup r?????? log logT(r,f)logr= limsup r?????? log log logM(r,f)logr.We also de???ne the hyper-lower-order as follows.??(f) = liminf r?????? log logT(r,f)logr= liminf r?????? log log logM(r,f)logr.In 1982, Bank and Laine [1] investigated the solutions of a di???erential equa-tion and obtained the following result.TheoremA. Let A(z) be a nonconstant polynomial of degree n, and let f 6= 0be a solution of the equation f

In this paper, Mond-Weir type duality results for a uncertainmultiobjective robust optimization problem are given under generalizedinvexity assumptions. Also, weak vector saddle-point theorems are ob-tained under convexity assumptions. 1. IntroductionConsider an uncertain multiobjective robust optimization problem:(MRP) minimize (f 1 (x),...,f l (x))subject to g j (x,v j )

In this paper, a finite difference scheme for a two-point boun-dary value problem of 2nth-order ordinary differential equations is pre-sented. The convergence and uniqueness of the solution for the schemeare proved by means of theories on matrix eigenvalues and norm. Nu-merical examples show that our method is very simple and effective, andthat this method can be used effectively for other types of boundary valueproblems. 1. IntroductionAssume that a 2nth-order (n ≥ 2) two-point boundary valve problem (BVP)is given in the formy (2n) = f(x)y +g(x), 0 < x < 1,y (2α) (0) = a α , α = 0,1,...,n−1,y (2α) (1) = b α ,(1.1)where f(x) and g(x) are continuous functions on the closed interval [0,1] anda α ,b α are all real constants. The problem (1.1) frequently occurs in structuralengineering, astrophysics and other branches of physical sciences. The problem(1.1) has a unique analytic solution for a fixed positive integer n ≥ 2 under thegeneral condition [5](1.2) f(x) 6= ( −1) n j 2n